Understanding Counterexamples in Mathematics: Disproving Statements with a Singular Example

Counterexample

A counterexample is a specific example that disproves a statement or a claim

A counterexample is a specific example that disproves a statement or a claim. It is used to show that a general statement is not always true by providing a singular situation where it fails.

To understand counterexamples, let’s consider an example:

Statement: “All triangles have equal side lengths.”

To disprove this statement, we need to find a single example of a triangle where the side lengths are not equal. Let’s consider the following triangle:

Triangle ABC, where:
– AB = 3 units
– BC = 4 units
– AC = 5 units

Here, we can see that the side lengths AB, BC, and AC are not equal. This example serves as a counterexample because it contradicts the statement that all triangles have equal side lengths.

By providing this specific triangle, we demonstrate that the statement is incorrect because we found an example where it does not hold true. The counterexample helps to highlight that the original claim is not universally valid for all possible scenarios.

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